Theorem REF  shows that \(\operatorname{CV}_{\mathrm {mod}}\)  (REF ), used with optimal partitioning, estimates the number of change-points consistently, under mild conditions.
The first condition in (REF ) is slightly stronger than the minimax rate for change-points to be detectable, which is \(\liminf _{n\rightarrow \infty } \underline{\lambda } \Delta _{(1)}^2 /(\sigma ^2 \log \overline{\lambda }) = \infty \) ; see [1]} and [2]}, [3]}, [4]} for related discussions. The second condition in (REF ) ensures that change-point locations are estimated accurately enough such that the costs caused by miss-estimating change-point locations are smaller than the costs of adding a false positive.
